The U.S. hunt for National Security Agency leaker Edward Snowden came to a boil Monday as the White House ripped into Hong Kong and China and issued warnings to Russia and Ecuador, where Mr. Snowden has sought asylum, sharply dialing up global pressure for his return to face espionage charges.

The case of Mr. Snowden, under federal indictment for stealing and leaking classified documents, has become a test of Washington's ability...